The reason is the Hand Limit rule is instantaneous. Actually, it's constantly instantaneous - if that makes sense.

For example, consider this case:
The Hand Limit 2 rule is in place at the start of your turn. Also, Draw 2 and Play 2 are in effect.
Therefore, everyone has 2 or less cards in their hand. For ease of example, let's assume everyone has two.
You Draw two cards. (For the time being, please either ignore the possibility that you may discard two cards, or assume that you choose not to.)
You Play the card: Everybody Gets One.
Once you complete this action, everyone else at the table has three cards and you have four.
Uh-oh! They're past the Hand Limit rule which is 'constantly instantaneous' and MUST discard one card.
Because it is your turn, you MAY discard two cards if you like.
Your turn continues.

This is no different from Andy's example:
The Hand Limit 2 rule is in place at the start of your turn. Also, Draw 2 and Play 2 are in effect.
You have two cards at the start of your turn
You Draw two cards.
Uh-oh! You're past the Hand Limit rule which is 'constantly instantaneous'.
Because it is your turn, you MAY discard two cards if you like.
Your turn continues.

So why include in the strict order 3) Follow Hand Limit, 4) Follow Keeper Limit.
First, as a reminder that once you have drawn and then played, your turn is coming to a close and you must now adhere to the Limit rules.
Second, so there is no debate about when your turn ends and the next person begins. I.e., if the rules were just 1) Draw, and 2) Play then we could debate, well does it become the next person's turn and then you follow Limit rules or do you follow the limit rules and then it's the next person's turn.

Does this settle the argument over whether there's a strict order? Maybe not. But I hope this scenario allows us to agree that Andy's you tube example does NOT mean you can follow the rules in any order. If anything his example is silent on the subject, though I would point out that during the course of the example he did follow the 'strict order' of play, he simply applied the 'constantly instantaneous' rule at the moment it affected him.

In the meantime, I will continue to take my turn by following the rules in order.